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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.3.79
Chapter 7, Problem 7.3.79

In Exercises 59–86, find the derivative of y with respect to the given independent variable.
79. y = θ sin(log₇ θ)

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Identify the function y = \(\theta\) \(\sin\)(\(\log\)_{7} \(\theta\)) and recognize that it is a product of two functions: u(\(\theta\)) = \(\theta\) and v(\(\theta\)) = \(\sin\)(\(\log\)_{7} \(\theta\)).
Apply the product rule for derivatives: \(\frac{dy}{d\theta}\) = u'(\(\theta\)) v(\(\theta\)) + u(\(\theta\)) v'(\(\theta\)).
Compute u'(\(\theta\)), the derivative of \(\theta\) with respect to \(\theta\), which is 1.
Find v'(\(\theta\)), the derivative of \(\sin\)(\(\log\)_{7} \(\theta\)). Use the chain rule: the derivative of \(\sin\)(x) is \(\cos\)(x), so multiply by the derivative of \(\log\)_{7} \(\theta\).
Recall that \(\log\)_{7} \(\theta\) = \(\frac{\ln \theta}{\ln 7}\), so its derivative is \(\frac{1}{\theta \ln 7}\). Combine these results to express v'(\(\theta\)) and then substitute back into the product rule formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative of Trigonometric Functions

Understanding how to differentiate trigonometric functions like sine is essential. The derivative of sin(u) with respect to x is cos(u) times the derivative of u, applying the chain rule. This allows us to handle compositions involving sine.
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Derivatives of Other Inverse Trigonometric Functions

Derivative of Logarithmic Functions with Arbitrary Bases

The derivative of log base a of x, log_a(x), can be expressed using natural logarithms as (1 / (x ln a)). Recognizing this helps differentiate expressions involving logarithms with bases other than e.
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Derivatives of General Logarithmic Functions

Product Rule for Differentiation

When differentiating a product of two functions, such as θ times sin(log₇ θ), the product rule states that the derivative is the first function times the derivative of the second plus the second function times the derivative of the first. This rule is crucial for correctly finding the derivative.
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The Product Rule
Related Practice
Textbook Question

25. First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of δ-gluconolactone into gluconic acid, for example,

dy/dt = -0.6y

when t is measured in hours. If there are 100 grams of δ-gluconolactone present when t=0, how many grams will be left after the first hour?

Textbook Question

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

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Textbook Question

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